Page 77 - Calculus Workbook For Dummies

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Chapter 5: Getting the Big Picture: Differentiation Basics

5. With your graphing calculator, graph both 6. Draw a function containing three points
the line y = - 4 x + 9 and the parabola where — for three different reasons — you
2
5
y = - x . You’ll see that they’re tangent at would not be able to determine the slope
the point (2, 1). and, thus, where you would not be able to
find a derivative.
2
a. What is the derivative of y = - x when
5
x = 2?
Solve It
b. On the parabola, how fast is y changing
compared to x when x = 2?
Solve It

The Handy-Dandy Difference Quotient

The difference quotient is the almost magical tool that gives us the slope of a curve at a
single point. To make a long story short, here’s what happens when you use the differ-
ence quotient. (If you want an excellent version of the long story, check out Calculus
For Dummies.) Look again at the figure in problem 3. You can’t get the slope of the
y 2 - y 1
parabola at (7,9) with the algebra slope formula, m = x 2 - x 1 n, because no matter
d
what other point on the parabola you use with (7,9) in the formula, you’ll get a slope
that’s steeper or less steep than the precise slope at (7,9).

But if your second point on the parabola is extremely close to (7,9) — like
9
` . 7 001 , .0029996j — your line would be almost exactly as steep as the tangent line.
The difference quotient gives the precise slope of the tangent line by sliding the
second point closer and closer to (7,9) until its distance from (7,9) is infinitely small.
Enough of this mumbo jumbo; now for the math. Here’s the definition of the derivative
based on the difference quotient:
f x + h - ^h f xh
^
x =
f l ^ h lim
h " 0 h
Like with most limit problems, plugging the arrow-number in at the beginning of a
0
difference quotient problem won’t help because that gives you . You have to do a
0
little algebraic mojo so that you can cancel the h and then plug in. (The techniques
from Chapter 4 also work here.)
Now for a difference quotient problem.